O What is the probability of randomly selecting 115 observations from a popula whose mean is 50 and whose standard deviation is 12.5 and finding that the less than 47?
O What is the probability of randomly selecting 115 observations from a popula whose mean is 50 and whose standard deviation is 12.5 and finding that the less than 47?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
![**Question:** What is the probability of randomly selecting 115 observations from a population whose mean is 50 and whose standard deviation is 12.5, and finding that the mean is less than 47?
**Explanation:**
This question involves calculating the probability using the normal distribution. To solve it, you need to use the concept of the sampling distribution of the sample mean.
1. **Identify the Given Information:**
- Population mean (μ) = 50
- Population standard deviation (σ) = 12.5
- Sample size (n) = 115
- Sample mean (M) = 47
2. **Calculate the Standard Error of the Mean (SEM):**
\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{12.5}{\sqrt{115}}
\]
3. **Find the Z-score:**
- Use the formula to determine the Z-score for the sample mean:
\[
Z = \frac{M - \mu}{\text{SEM}}
\]
4. **Determine the Probability:**
- Use the Z-score to find the probability from the Z-table or a standard normal distribution calculator.
By following these steps, you can calculate the probability of the sample mean being less than 47.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d8d143f-a211-410b-93d7-c25b2730a251%2F6bc95498-4929-4adb-9b1a-1dde06effe32%2F9h0frkn_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:** What is the probability of randomly selecting 115 observations from a population whose mean is 50 and whose standard deviation is 12.5, and finding that the mean is less than 47?
**Explanation:**
This question involves calculating the probability using the normal distribution. To solve it, you need to use the concept of the sampling distribution of the sample mean.
1. **Identify the Given Information:**
- Population mean (μ) = 50
- Population standard deviation (σ) = 12.5
- Sample size (n) = 115
- Sample mean (M) = 47
2. **Calculate the Standard Error of the Mean (SEM):**
\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{12.5}{\sqrt{115}}
\]
3. **Find the Z-score:**
- Use the formula to determine the Z-score for the sample mean:
\[
Z = \frac{M - \mu}{\text{SEM}}
\]
4. **Determine the Probability:**
- Use the Z-score to find the probability from the Z-table or a standard normal distribution calculator.
By following these steps, you can calculate the probability of the sample mean being less than 47.
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